Life is full of ironies and contradictions. Case in point: I write for two competitions (Who Wants to Be a Mathematician, and MATHCOUNTS), but I never have participated in a math competition myself. I hadn’t even heard of the Putnam until graduate school. It gets better: while I now write for and have helped students prep for competitions (not the ones I just mentioned, don’t worry. I know what a conflict of interest is.), “the jury is still out” as to whether or not I think they’re good ideas.
I have personal experience with competition, just not math competitions. From the age of five to nineteen I studied (classical) piano. When I was closer to five than to nineteen, a competition meant I wore fancy black clothing, got to spend hours in the car with my father, only 10 minutes playing in front of a stranger, and then the day would end somehow with me consuming ice cream.
But as I got older, I realized only one person wins a (piano) competition. The pressure to win, from any and all sources, made competition a high stakes game (though an increasingly not-fun game). When competition is more than just a trophy, but also a step towards “the next level” stakes are even higher. The psychology and pressure and amount of time involved can become so negative, that–like me–one day you just quit playing.
So why do I write for math competitions? Part of it is Joe Gallian’s Project NExT line of “Just say ‘yes’”—a piece of advice I suspect may be about as healthy as competitions. Still, I have been known to say “no”: most notably, I’ve refused offers to contribute to the AMC 10/12. Perhaps I’m deluding myself, but I feel the competitions I write for are the equivalent of the “ice cream afterwards” experiences. The stakes, compared to the AMC 10/12, or the ARML, or the AIME, or the Olympiads, or the Putnam, are much lower for these kids.
Still here are two stories from my friends regarding competitions that I use as my angel and demon on my shoulder whenever I think about writing for one more year:
(1) A friend/can’t-be-summarized of mine (and one of the most well-rounded mathematicians, let alone most beautiful minds, I’ve ever met) said that he took the Putnam exactly once, as a freshman, earning low double-digits. I said, “That’s really impressive! Why did you stop?” He said, “Because I learned it was out of 120 points.” He argued a 0/120 would have been less insulting. That he’d feel better knowing he was totally out of his league than to feel like he was good at math and had spent all this time in seminars prepping and only earn about 10%.
Now luckily for all of us, this friend was NOT turned off of math because of this experience. Just competitions. Still I can’t help but wonder if those younger or less mature than he was can see the difference. It’s a great fear of mine that even for “my” competitions something similar could be happening. One of the organizations has flat-out said their goal is to make sure that everyone who participates gets at least some questions right. But if these students are putting in the time to prepare, and think they’re really good at math, and they’re being TOLD they’re really good at math, and realistically they probably legitimately are good at math, but they get a 10% at this competition, won’t history repeat itself?
(2) Another friend/can’t-be-quantified of mine (who writes for the “not ice-cream” style competitions) has said this about why he writes. Smart kids first need to be challenged, among other reasons so they don’t get bored and move on to a subject other than math. Realistically, they are in part flocking to competitions because their curiosities aren’t piqued from their standard curriculum and we as a mathematical community don’t want to suffer that loss of talent. Prepping for these competitions, which start with pedestrian topics but take them in remarkably creative directions, addresses that.
But just as crucially, these smart kids are with VERY few exceptions sincerely and severely lacking in humility. I’ve witnessed this first-hand in local, state and national outreach programs for really precocious kids, and even at some of the colleges where I’ve taught. I’ve had a 12-year old “man-splain” something to me (and the 12-year old was just wrong). These kids are as ugly as any competitors I’ve ever encountered, myself included. They think because they have been the most mathematically gifted in the room for most of their lives that—no matter what—they are the most intellectually gifted in every room. My friend/unquantifiable basically argues that competitions can show them they still have something left to learn.
Regardless of our personal opinions of or experiences with competitions we still can all benefit from examining competition problems. Maybe we can use them as inspiration for “cool” class examples, structured homework exercises or group work activities. As educators we have much to gain from competitions including
*Learning what’s en vogue. In music competitions, you can either follow the trend or not. I’m fine bucking norms without criticizing them: for a time, Chopin’s “Fantasie Impromptu” was on many recital programs, which incentivized me to learn Chopin’s first ballade (later popularized by The Pianist, which made me learn Chopin’s fourth ballade). Math competitions are similar. Certain combinatorics tricks are really “in” right now, as are certain factoring tricks. Maybe it’s just my competitions, but overall geometry is emphasized less than it was in the same events 30+ years ago. This is good information to have. It helps us know what students may know coming in to our courses. It helps us see what others in the greater mathematical community are stressing.
*Seeing what “top” students are capable of. One thing I have not yet mentioned is that competitions can be inspirational. Musically, you literally hear new things and see new programs. You’d think “Wow…who would pair Bach’s Toccata in g with Mozart’s Fugue in C?! That sounds amazing!” Math competitions allow us to see what can be done in an hour, 10 minutes, 30 seconds…if you put your mind to it. Competitions give us lofty goals for ourselves and our students. I, at least, think “God, that’s a really avant-garde application of [insert basic, “boring” to students, theorem]. That’ll at least wake them up!”
*Challenging ourselves to be better. I look at a lot of old competitions while writing for the current ones. And when I think I’m really “borrowing” an idea, I’ll cite, and I have been told on occasion to change more of the set-up. But what that whole process forces me to do is an extremely detailed post-mortem. I am forced to analyze “What is the point of this? What is the trick? Is there more than one trick? How easy is this to manipulate to other situations? What if you increase the number of variables? Or allow rationals instead of integers?” Passing on any of this thought-process to students is also beneficial. Having them see how a problem does (or doesn’t) change by altering certain quantities is a useful skill. It gets THEM thinking too.